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\begin{document}

\title{List of noteworthy papers\\ Est. 9/14/11}
\author{Leonid Petrov}
\maketitle



\section{Differential posets, Kerov's operators}

\subsection{Down-up Algebras // Benkart, G. and Roby, T. // 1998}
\four

\aurl{http://arxiv.org/abs/math/9803159v1}

\abst{The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-up algebras. We show that down-up algebras exhibit many of the important features of the universal enveloping algebra $U(sl2)$ of the Lie algebra $sl2$ including a Poincar\'e-Birkhoff-Witt type basis and a well-behaved representation theory. We investigate the structure and representations of down-up algebras and focus especially on Verma modules, highest weight representations, and category $\mathcal O$ modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets.}

\comment Here the authors consider algebras with relations coming from commutation relations of ``up'' and ``down'' operators on posets. There are differential posets, $r$-differential posets, and more posets. The posets are just motivation to study general picture of down-up algebras. 

Representations of the down-up algebras are studied, and more.

\subsection{Three papers by Jan Snellman on composition posets // 2002--2003}

\four

\aurl{http://arxiv.org/abs/math/0201083}

\aurl{http://arxiv.org/abs/math/0309458}

\aurl{http://arxiv.org/abs/math/0505262}

\comment There are various ways to turn the set of compositions into a poset. Various ways are given in these papers and in \ref{stanley_comp_pos}, \ref{bbd_comp_pos} also. Saturated chains are studied (= dimension functions), and in some cases algebraic structures (e.g. quasi-symmetric functions) are involved.

\subsection{An Analogue of Young’s Lattice for Compositions // Anders Bjorner, Richard P. Stanley // 2005}
\label{stanley_comp_pos}

\two

\aurl{http://arxiv.org/abs/math/0508043}

\comment Fundamental quasi-symmetric functions and compositions. Seems likely that this is the graph of zigzag diagrams. Gnedin and Olshanski described its boundary in \cite{GnedinIntern.Math.ResearchNotices2006Art.ID5196839pp.}.

\subsection{Standard paths in the composition poset // Bergeron, Bousquet-Melou, Dulucq // 1995}
\label{bbd_comp_pos}

\zero

\refer{Ann. Sci. Math. Quebec, 19(2):139–151, 1995.}

\subsection{Mixing time for a random walk on rooted trees // Jason Fulman // 2008}

\five

\comment This is a paper on Kerov-type operators on rooted trees. Fulman as usual deals with spectral structure of down/up Markov chains. However, what is the limiting object? The boundary of the graph? What will be the limiting Markov process?

Moreover, this model is Kerov-type, so it must have a Meixner-type mixed jump process. Does it have a limit? At least I know how to diagonalize its generator.

\subsection{Work of Proctor}

\zero;
\four

(See the end of Stanley's Variations on Differential Posets)

Here Proctor uses $\slf(2)$ operators on posets (however, these posets are finite). Some of his work is inacessible, but you can find some.

The most interesting (and not listed by Stanley) seems to be ``Solution of a sperner conjecture of Stanley with a construction of Gelfand'', but it is also inacessible.



\section{Random Matrices}

\subsection{Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product // S. Belinschi, B. Collins, I. Nechita // 2010}
\zero

\aurl{http://arxiv.org/abs/1008.3099v2}

\abst{For a parameter $t\in (0,1)$ and an integer $n$, we choose at random a vector subspace $V_n\subset \mathbb{C}^k\otimes\mathbb{C}^n$ of dimension $N\sim tnk$. 
We exhibit a cone that partitions $\R_+^k$ into two connected components, such that, for any sequence in the complement of the cone, the probability that it occurs as the set of singular values of some vector of $V_n$ is either 0 or 1 as $n\to\infty$. 
Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high multiplicity.}

\comment Chris King: ``Proposition 5.1 is the key result which I would like to understand -- Chris''. 

The method here is free probability, which we don't understand. The paper deals with singular values of large random matrices, and establishes a law of large numbers.



\section{Tiling models}


\subsection{Home page of Casey Warmbrand}

\aurl{http://math.arizona.edu/~caseyw/}

\comment Has nice slides about the Aztec diamond.



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